Scaling laws for multi-object spectrographs: empirical relationships and the photonic advantage of CAWSMOS

Abstract

Conventional multi-object spectrographs (MOS) incur masses and costs that scale unfavourably with channel count~$N_\mathrm{ch}$ and resolving power~$\mathcal{R}$. A dataset of eight fibre-fed MOS-VIS instruments (1995--2024) is compiled and analysed. Ordinary least-squares fitting yields a mass function $M \propto N_\mathrm{ch}^{0.78\pm0.28}$; the resolving-power exponent is empirically consistent with zero ($\gamma = 0.20\pm0.37$) but is predicted analytically to lie in $\mathcal{R}^{0.8\text{--}2.0}$. Leave-one-out cross-validation confirms sub-linear channel-count scaling ($\beta\in[0.59,1.13]$, mean 0.79), with HERMES and AAOmega -- sharing the same telescope and $N_\mathrm{ch}=392$ across a factor of~7 in $\mathcal{R}$ at essentially equal estimated mass -- providing a direct empirical confirmation that $\gamma\approx 0$. Cost per channel decreases as $N_\mathrm{ch}^{-0.22\text{ to }-0.53}$ for all assumed cost--mass exponents $k\in[0.6,1.0]$, demonstrating a genuine economy of scale. The photonic integrated-circuit (PIC) approach embodied by the arrayed waveguide grating (AWG)-based PAWS demonstrator and the proposed CAWSMOS instrument decouples dispersive element size from $N_\mathrm{ch}$ and $\mathcal{R}$, with chip-area scaling as $\mathcal{R}^{1\text{--}1.5}$ instead of $\mathcal{R}^{3}$. Applied to the Wide-field Spectroscopic Telescope (WST), the model predicts $\sim$140 tonnes for a conventional NIR spectrograph system against $\sim$200 kg for a CAWSMOS equivalent, with projected costs of EUR 300--700M conventional versus EUR 40--120M photonic ($6$--$15\times$ reduction). AWG-based photonics are identified as a strategially important enabling technology for WST.